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Theorem opelvvg 4548
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
opelvvg  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )

Proof of Theorem opelvvg
StepHypRef Expression
1 elex 2668 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2668 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 opelxpi 4531 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
41, 2, 3syl2an 285 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1463   _Vcvv 2657   <.cop 3496    X. cxp 4497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950  df-xp 4505
This theorem is referenced by:  relsnopg  4603
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